3.273 \(\int \frac{a+b x^2}{(-c+d x)^{3/2} (c+d x)^{3/2}} \, dx\)

Optimal. Leaf size=63 \[ \frac{2 b \tanh ^{-1}\left (\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right )}{d^3}-\frac{x \left (\frac{a}{c^2}+\frac{b}{d^2}\right )}{\sqrt{d x-c} \sqrt{c+d x}} \]

[Out]

-(((a/c^2 + b/d^2)*x)/(Sqrt[-c + d*x]*Sqrt[c + d*x])) + (2*b*ArcTanh[Sqrt[-c + d
*x]/Sqrt[c + d*x]])/d^3

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Rubi [A]  time = 0.114977, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ \frac{2 b \tanh ^{-1}\left (\frac{\sqrt{d x-c}}{\sqrt{c+d x}}\right )}{d^3}-\frac{x \left (\frac{a}{c^2}+\frac{b}{d^2}\right )}{\sqrt{d x-c} \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)/((-c + d*x)^(3/2)*(c + d*x)^(3/2)),x]

[Out]

-(((a/c^2 + b/d^2)*x)/(Sqrt[-c + d*x]*Sqrt[c + d*x])) + (2*b*ArcTanh[Sqrt[-c + d
*x]/Sqrt[c + d*x]])/d^3

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Rubi in Sympy [A]  time = 21.0818, size = 94, normalized size = 1.49 \[ - \frac{a x}{c^{2} \sqrt{- c + d x} \sqrt{c + d x}} - \frac{b c}{d^{3} \sqrt{- c + d x} \sqrt{c + d x}} - \frac{b \sqrt{- c + d x}}{d^{3} \sqrt{c + d x}} + \frac{2 b \operatorname{atanh}{\left (\frac{\sqrt{- c + d x}}{\sqrt{c + d x}} \right )}}{d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)/(d*x-c)**(3/2)/(d*x+c)**(3/2),x)

[Out]

-a*x/(c**2*sqrt(-c + d*x)*sqrt(c + d*x)) - b*c/(d**3*sqrt(-c + d*x)*sqrt(c + d*x
)) - b*sqrt(-c + d*x)/(d**3*sqrt(c + d*x)) + 2*b*atanh(sqrt(-c + d*x)/sqrt(c + d
*x))/d**3

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Mathematica [A]  time = 0.0934242, size = 100, normalized size = 1.59 \[ \frac{d x \sqrt{d x-c} \sqrt{c+d x} \left (a d^2+b c^2\right )+b c^2 \left (c^2-d^2 x^2\right ) \log \left (\sqrt{d x-c} \sqrt{c+d x}+d x\right )}{c^2 d^3 (c-d x) (c+d x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)/((-c + d*x)^(3/2)*(c + d*x)^(3/2)),x]

[Out]

(d*(b*c^2 + a*d^2)*x*Sqrt[-c + d*x]*Sqrt[c + d*x] + b*c^2*(c^2 - d^2*x^2)*Log[d*
x + Sqrt[-c + d*x]*Sqrt[c + d*x]])/(c^2*d^3*(c - d*x)*(c + d*x))

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Maple [C]  time = 0.027, size = 160, normalized size = 2.5 \[{\frac{{\it csgn} \left ( d \right ) }{{c}^{2}{d}^{3}} \left ( \ln \left ( \left ({\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-{c}^{2}}+dx \right ){\it csgn} \left ( d \right ) \right ){x}^{2}b{c}^{2}{d}^{2}-ax\sqrt{{d}^{2}{x}^{2}-{c}^{2}}{d}^{3}{\it csgn} \left ( d \right ) -b{c}^{2}x\sqrt{{d}^{2}{x}^{2}-{c}^{2}}{\it csgn} \left ( d \right ) d-b{c}^{4}\ln \left ( \left ({\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-{c}^{2}}+dx \right ){\it csgn} \left ( d \right ) \right ) \right ){\frac{1}{\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt{dx-c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)/(d*x-c)^(3/2)/(d*x+c)^(3/2),x)

[Out]

(ln((csgn(d)*(d^2*x^2-c^2)^(1/2)+d*x)*csgn(d))*x^2*b*c^2*d^2-a*x*(d^2*x^2-c^2)^(
1/2)*d^3*csgn(d)-b*c^2*x*(d^2*x^2-c^2)^(1/2)*csgn(d)*d-b*c^4*ln((csgn(d)*(d^2*x^
2-c^2)^(1/2)+d*x)*csgn(d)))*csgn(d)/(d^2*x^2-c^2)^(1/2)/c^2/d^3/(d*x+c)^(1/2)/(d
*x-c)^(1/2)

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Maxima [A]  time = 1.41802, size = 115, normalized size = 1.83 \[ -\frac{a x}{\sqrt{d^{2} x^{2} - c^{2}} c^{2}} - \frac{b x}{\sqrt{d^{2} x^{2} - c^{2}} d^{2}} + \frac{b \log \left (2 \, d^{2} x + 2 \, \sqrt{d^{2} x^{2} - c^{2}} \sqrt{d^{2}}\right )}{\sqrt{d^{2}} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)/((d*x + c)^(3/2)*(d*x - c)^(3/2)),x, algorithm="maxima")

[Out]

-a*x/(sqrt(d^2*x^2 - c^2)*c^2) - b*x/(sqrt(d^2*x^2 - c^2)*d^2) + b*log(2*d^2*x +
2*sqrt(d^2*x^2 - c^2)*sqrt(d^2))/(sqrt(d^2)*d^2)

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Fricas [A]  time = 0.237107, size = 153, normalized size = 2.43 \[ \frac{b c^{2} + a d^{2} -{\left (b d^{2} x^{2} - \sqrt{d x + c} \sqrt{d x - c} b d x - b c^{2}\right )} \log \left (-d x + \sqrt{d x + c} \sqrt{d x - c}\right )}{d^{5} x^{2} - \sqrt{d x + c} \sqrt{d x - c} d^{4} x - c^{2} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)/((d*x + c)^(3/2)*(d*x - c)^(3/2)),x, algorithm="fricas")

[Out]

(b*c^2 + a*d^2 - (b*d^2*x^2 - sqrt(d*x + c)*sqrt(d*x - c)*b*d*x - b*c^2)*log(-d*
x + sqrt(d*x + c)*sqrt(d*x - c)))/(d^5*x^2 - sqrt(d*x + c)*sqrt(d*x - c)*d^4*x -
c^2*d^3)

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Sympy [A]  time = 70.1161, size = 182, normalized size = 2.89 \[ a \left (- \frac{{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4}, 1 & \frac{1}{2}, \frac{3}{2}, 2 \\\frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2}, 2 & 0 \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} c^{2} d} + \frac{i{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1 & \\\frac{1}{4}, \frac{3}{4} & - \frac{1}{2}, 0, 1, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} c^{2} d}\right ) + b \left (\frac{{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{4} & - \frac{1}{2}, \frac{1}{2}, 1, 1 \\- \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 1, 0 & \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} d^{3}} + \frac{i{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{3}{2}, -1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 1 & \\- \frac{3}{4}, - \frac{1}{4} & - \frac{3}{2}, -1, 0, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} d^{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)/(d*x-c)**(3/2)/(d*x+c)**(3/2),x)

[Out]

a*(-meijerg(((3/4, 5/4, 1), (1/2, 3/2, 2)), ((3/4, 1, 5/4, 3/2, 2), (0,)), c**2/
(d**2*x**2))/(2*pi**(3/2)*c**2*d) + I*meijerg(((-1/2, 0, 1/4, 1/2, 3/4, 1), ()),
((1/4, 3/4), (-1/2, 0, 1, 0)), c**2*exp_polar(2*I*pi)/(d**2*x**2))/(2*pi**(3/2)
*c**2*d)) + b*(meijerg(((-1/4, 1/4), (-1/2, 1/2, 1, 1)), ((-1/4, 0, 1/4, 1/2, 1,
0), ()), c**2/(d**2*x**2))/(2*pi**(3/2)*d**3) + I*meijerg(((-3/2, -1, -3/4, -1/
2, -1/4, 1), ()), ((-3/4, -1/4), (-3/2, -1, 0, 0)), c**2*exp_polar(2*I*pi)/(d**2
*x**2))/(2*pi**(3/2)*d**3))

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GIAC/XCAS [A]  time = 0.231704, size = 153, normalized size = 2.43 \[ -\frac{b{\rm ln}\left ({\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2}\right )}{d^{3}} - \frac{2 \,{\left (b c^{2} + a d^{2}\right )}}{{\left ({\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2} + 2 \, c\right )} c d^{3}} - \frac{{\left (b c^{2} d^{3} + a d^{5}\right )} \sqrt{d x + c}}{2 \, \sqrt{d x - c} c^{2} d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)/((d*x + c)^(3/2)*(d*x - c)^(3/2)),x, algorithm="giac")

[Out]

-b*ln((sqrt(d*x + c) - sqrt(d*x - c))^2)/d^3 - 2*(b*c^2 + a*d^2)/(((sqrt(d*x + c
) - sqrt(d*x - c))^2 + 2*c)*c*d^3) - 1/2*(b*c^2*d^3 + a*d^5)*sqrt(d*x + c)/(sqrt
(d*x - c)*c^2*d^6)